Estimating the probability that a given vector is in the convex hull of a random sample
For a d-dimensional random vector X, let pn,X (θ ) be the probability that the convex hull of n independent copies of X contains a given point θ. We provide several sharp inequalities regarding pn,X (θ ) and NX (θ ) denoting the smallest n for which pn,X (θ ) ≥ 1/2. As a main result, we derive the t...
| Main Authors: | , , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
Springer
2023
|
| _version_ | 1826310460158246912 |
|---|---|
| author | Hayakawa, S Lyons, T Oberhauser, H |
| author_facet | Hayakawa, S Lyons, T Oberhauser, H |
| author_sort | Hayakawa, S |
| collection | OXFORD |
| description | For a d-dimensional random vector X, let pn,X (θ ) be the probability that the convex hull of n independent copies of X contains a given point θ. We provide several sharp inequalities regarding pn,X (θ ) and NX (θ ) denoting the smallest n for which pn,X (θ ) ≥ 1/2. As a main result, we derive the totally general inequality 1/2 ≤ αX (θ )NX (θ ) ≤ 3d + 1, where αX (θ ) (a.k.a. the Tukey depth) is the minimum probability that X is in a fixed closed halfspace containing the point θ. We also show several applications of our general results: one is a moment-based bound on NX (E[X]), which is an important quantity in randomized approaches to cubature construction or measure reduction problem. Another application is the determination of the canonical convex body included in a random convex polytope given by independent copies of X, where our combinatorial approach allows us to generalize existing results in random matrix community significantly. |
| first_indexed | 2024-03-07T07:51:24Z |
| format | Journal article |
| id | oxford-uuid:f29787cf-23d6-4e3c-94cd-ec9f0349cda9 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:51:24Z |
| publishDate | 2023 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:f29787cf-23d6-4e3c-94cd-ec9f0349cda92023-07-13T09:48:45ZEstimating the probability that a given vector is in the convex hull of a random sampleJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:f29787cf-23d6-4e3c-94cd-ec9f0349cda9EnglishSymplectic ElementsSpringer2023Hayakawa, SLyons, TOberhauser, HFor a d-dimensional random vector X, let pn,X (θ ) be the probability that the convex hull of n independent copies of X contains a given point θ. We provide several sharp inequalities regarding pn,X (θ ) and NX (θ ) denoting the smallest n for which pn,X (θ ) ≥ 1/2. As a main result, we derive the totally general inequality 1/2 ≤ αX (θ )NX (θ ) ≤ 3d + 1, where αX (θ ) (a.k.a. the Tukey depth) is the minimum probability that X is in a fixed closed halfspace containing the point θ. We also show several applications of our general results: one is a moment-based bound on NX (E[X]), which is an important quantity in randomized approaches to cubature construction or measure reduction problem. Another application is the determination of the canonical convex body included in a random convex polytope given by independent copies of X, where our combinatorial approach allows us to generalize existing results in random matrix community significantly. |
| spellingShingle | Hayakawa, S Lyons, T Oberhauser, H Estimating the probability that a given vector is in the convex hull of a random sample |
| title | Estimating the probability that a given vector is in the convex hull of a random sample |
| title_full | Estimating the probability that a given vector is in the convex hull of a random sample |
| title_fullStr | Estimating the probability that a given vector is in the convex hull of a random sample |
| title_full_unstemmed | Estimating the probability that a given vector is in the convex hull of a random sample |
| title_short | Estimating the probability that a given vector is in the convex hull of a random sample |
| title_sort | estimating the probability that a given vector is in the convex hull of a random sample |
| work_keys_str_mv | AT hayakawas estimatingtheprobabilitythatagivenvectorisintheconvexhullofarandomsample AT lyonst estimatingtheprobabilitythatagivenvectorisintheconvexhullofarandomsample AT oberhauserh estimatingtheprobabilitythatagivenvectorisintheconvexhullofarandomsample |